X3652

KS(CARNOT TRIANGLE)

Quim Castellsaguer defines the Carnot triangle as the triangle formed by the circumcenters of the triangles BCH, CAH, ABH, where H denotes the orthocenter. The Carnot triangle is also known as the Johnson triangle.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 30, 40 }: Central Line L(30, 40) { 5, 79 }: Central Line L(5, 79) { 21, 104 }: Central Line L(21, 104) { 500, 846 }: Central Line L(500, 846) { 12, 1727 }: Central Line L(12, 1727) { 140, 1768 }: Central Line L(140, 1768) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_40 = BEVAN POINT (0.4286, -0.0292) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_500 = ORTHOCENTER OF THE INCENTRAL TRIANGLE (0.5577, 0.1973) X_846 = 4th SHARYGIN POINT (0.5437, 0.1969) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1727 = MIMOSA TRANSFORM OF X(80) (1.2736, 0.8828) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_1768 = MIMOSA TRANSFORM OF X(521) (1.1429, -0.7582) X_3652 = KS(CARNOT TRIANGLE) (0.5146, 0.1959) A B C X30 X40 X5 X79 X21 X104 X500 X846 X12 X1727 X140 X1768 X3652 (a, b, c) = (5, 6, 7) | AB = 1

X3652 (KS(CARNOT TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{6} + a^{4}(bc - 3b^{2} - 3c^{2}) + a^{3}bc(b + c) + 3a^{2}(b^{4} + c^{4}) - abc(b + c)(b - c)^{2} - (b^{2} - c^{2})^{2}(b^{2} + bc + c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.058829177291\)
▸ Linear Combinations1 combination

  • X3652 = \((2r + 3R)\) X1 - \(9R\) X2 - \((4r - 3R)\) X3

▸ Incident Central Lines10 lines

{5, 79}, {12, 1727}, {21, 104}, {30, 40}, {140, 1768}, {500, 846}, {758, 1482}, {942, 1776}, {952, 3065}, {3219, 3579}